the genus of regular languages and other ideas from low ...deloup/topcomp2016/slides/genus.pdf ·...

A short overview of topology and languages interactionsIntroduction

Crash course on regular languagesGenus of a regular language

Genus and size

The genus of regular languages and other ideasfrom low-dimensional topology

Florian Deloup

Institut de Mathematiques de Toulouse, France

June 21, 2016

Florian Deloup The genus of regular languages and other ideas from low-dimensional topology

Genus and size

Joint work with Guillaume Bonfante.

1) The genus of regular languages, 2012. Math. Str. ComputerSc., 2016.

2) The decidability of language genus computation, 2016.Available on ArXiv.

Genus and size

Joint work with Guillaume Bonfante.

1) The genus of regular languages, 2012. Math. Str. ComputerSc., 2016.

2) The decidability of language genus computation, 2016.Available on ArXiv.

Genus and size

What I won’t talk about in this talk

Topology =⇒ Languages (as tool to study topology): languages astopological invariants

- Fundamentalgroup of a topologicalspace, languages (Poincare,1895, “Analysis situs” paper,also Riemann and Klein)

- Knots: encodingReidemeister moves (1927)yields language(s). Particular cases: quandles, Wirtingerpresentation of the fundamental group of the complement of aknot.

Genus and size

What I will talk about in this talk

Languages =⇒ Topology (as a tool to study languages): topologyas a language invariant

This talk: language invariants from low-dimensional topology.

Genus and size

”Moore’s Law”

Moore’s ”Law” (1960s)

The number of transistors in a dense integrated circuit doublesevery two years.

Correction to Moore’s ”Law” (2005)

Moore’s Law has to end.

Reason invoked: physical limit of matter processing.Shape and space organization become central =⇒Low-dimensional topology =⇒ Invariants of Languages fromlow-dimensional topology

Genus and size

”Moore’s Law”

Genus and size

”Moore’s Law”

Reason invoked: physical limit of matter processing.

Shape and space organization become central =⇒Low-dimensional topology =⇒ Invariants of Languages fromlow-dimensional topology

Genus and size

”Moore’s Law”

Genus and size

Regular languages as computationsAutomataMinimal automaton

Regular languages

Set-up:- the class RegA of regular languages on a finite alphabet A .- the class DFAA of deterministic finite automata on A .

Working-out definition: a regular language L on alphabet A is asubset of A ∗, starting from a subset of A and recursivelycomputed by a finite number of the familiar 3 operations:

- Union of two languages:(L, L′) 7→ L ∪ L′ = {w ∈ A∗ | w ∈ L, or w ∈ L′}.- Composition of two languages:(L, L′) 7→ LL′ = {ww ′ | w ∈ L,w ′ ∈ L′}.- Star operation: L 7→ L∗ =

⋃n≥0 Ln

Genus and size

Regular languages

⋃n≥0 Ln

Genus and size

Regular languages

⋃n≥0 Ln

Genus and size

Automata

An automaton is a decorated directed (multi)graph.

Genus and size

Automata

An automaton is a decorated directed (multi)graph.

Genus and size

Automata

Decoration:- label each directed edge (transition) by a letter of the alphabetA .

- distinguish special states: one initial state, one subset of finalstates.Pictorial convention for initial and final states:

initial,

final final

Genus and size

Automata

Decoration:- label each directed edge (transition) by a letter of the alphabetA .

- distinguish special states: one initial state, one subset of finalstates.Pictorial convention for initial and final states:

initial,

final final

Genus and size

Deterministic automaton

The automaton is deterministic if there is at most one transitionlabelled by a given letter.

Genus and size

Deterministic automaton

The automaton is deterministic if there is at most one transitionlabelled by a given letter.

Genus and size

Kleene

Let A ∈ DFA. The language L(A) computed by A is the set of allwords w ∈ A ∗ read from (the sequence of labels of) a pathstarting at the initial state and ending at some final state of A.

Theorem

The assignment A 7→ L(A) defines a surjective mapDFAA → RegA .

Genus and size

In the words of the topologists

Challenge: define “quantum invariants” of L = L(A) (beyond thesize of L), locally computable from a picture of any automaton A

computing L. The computation from two equivalent automatashould give the same invariant.

Why is it a challenge ? Nonlocal nature of the computation: twoautomata can be nonlocally equivalent.

Genus and size

The simplest invariant of language.

Definition

The size |L| of a language L is the smallest number of statesrequired to produce a deterministic automaton A computing L:

|L| = min{|A| | A ∈ DFA, L(A) = L}.

Theorem (Myhill-Nerode, 1950s)

Let L be a regular language. There is a unique automatonA ∈ DFA such that L(A) = L with number |A| of states equal to|L|.

Such an automaton is called the minimal automaton of L.

Genus and size

Definition

|L| = min{|A| | A ∈ DFA, L(A) = L}.

Genus and size

Definition

|L| = min{|A| | A ∈ DFA, L(A) = L}.

Genus and size

Definition

|L| = min{|A| | A ∈ DFA, L(A) = L}.

Genus and size

Classification of closed oriented surfacesEmbeddings into a closed oriented surfaceGenus of a regular language

Classification of closed oriented surfaces

Theorem (first stated 1850, proved 1920): The topological typeof a closed oriented surface Σ is determined by one natural numberg(Σ) ∈ N.

S0 S1 S2 S3

The genus is the number of “handles” required to produce thesurface Σ from the sphere.

The genus g(Σ) of Σ is the maximal number of mutually disjointsimple closed curves C1, . . . ,Cg such that the complementΣ− (C1 ∪ · · · ∪ Cg ) remains connected.

Genus and size

S0 S1 S2 S3

Genus and size

S0 S1 S2 S3

Genus and size

Examples

genus = 0 genus = 1 genus = 2 · · · · · ·

Genus and size

Embedding an automaton into a closed oriented surface

An embedding of a graph is essentially a “drawing of the graphwithout crossings of the edges”.

Definition

An embedding of a graph G = (E ,V ) into a closed oriented surfaceΣ is a map ϕ : (E ,V )→ Σ sending injectively vertices to points,sending edges to simple arcs in Σ such that ϕ(∂e) = ∂ϕ(e) for anyedge e ∈ E , ϕ(e) ∩ ϕ(e ′) = ϕ(∂e) ∩ ϕ(∂e ′) for any pair e, e ′ ∈ E .

Genus and size

Embedding an automaton into a closed oriented surface

An embedding of a graph is essentially a “drawing of the graphwithout crossings of the edges”.

Definition

An embedding of a graph G = (E ,V ) into a closed oriented surfaceΣ is a map ϕ : (E ,V )→ Σ sending injectively vertices to points,sending edges to simple arcs in Σ such that ϕ(∂e) = ∂ϕ(e) for anyedge e ∈ E , ϕ(e) ∩ ϕ(e ′) = ϕ(∂e) ∩ ϕ(∂e ′) for any pair e, e ′ ∈ E .

Genus and size

Example. The ”Utility Graph” (complete bipartite K3,3)

is not embeddable in the sphere (Kuratowski).

However, K3,3

embeds into a torus (genus 1).

Genus and size

Example. The ”Utility Graph” (complete bipartite K3,3)

is not embeddable in the sphere (Kuratowski). However, K3,3

embeds into a torus (genus 1).

Genus and size

Genus of a regular language

Definition

Let L be a regular language. The genus g(L) is defined as

g(L) = min{g(A) | A ∈ DFA, L(A) = L}.

If g(L) = 0, then L is said to be planar.

Remark: the definition makes sense because any graph embedsinto some closed oriented surface.

Genus and size

Genus of a regular language

Definition

Let L be a regular language. The genus g(L) is defined as

g(L) = min{g(A) | A ∈ DFA, L(A) = L}.

If g(L) = 0, then L is said to be planar.

Remark: the definition makes sense because any graph embedsinto some closed oriented surface.

Genus and size

Topological sizeComputabilityGenus growthHierarchies of languagesDirected EmulatorsSome recent speculations

Recall: the simplest invariant of a regular language L is its size

|L| = min{|A| | A ∈ DFA, L(A) = L}.

Question: relation between the genus and the size of a language ?

Genus and size

Recall: the simplest invariant of a regular language L is its size

|L| = min{|A| | A ∈ DFA, L(A) = L}.

Question: relation between the genus and the size of a language ?

Genus and size

Basic observation: the automaton A for which a minimalembedding (with minimal genus) is realized may not be theminimal automaton.

Alphabet: A = {0, 1, 2}Morphism: ϕ : A ∗ → Z/5Z defined by ϕ(aw) = ϕ(a) + ϕ(w) forany a ∈ A , w ∈ A ∗.Language: L = {w ∈ A ∗ | ϕ(w) = 0 mod 5}

Figure: Left: the minimal DFA in the sense of Myhill-Nerode. Right: aDFA of minimal genus (planar) recognizing the same language.

Genus and size

Basic observation: the automaton A for which a minimalembedding (with minimal genus) is realized may not be theminimal automaton.Alphabet: A = {0, 1, 2}Morphism: ϕ : A ∗ → Z/5Z defined by ϕ(aw) = ϕ(a) + ϕ(w) forany a ∈ A , w ∈ A ∗.Language: L = {w ∈ A ∗ | ϕ(w) = 0 mod 5}

Genus and size

Basic observation: the automaton A for which a minimalembedding (with minimal genus) is realized may not be theminimal automaton.Alphabet: A = {0, 1, 2}Morphism: ϕ : A ∗ → Z/5Z defined by ϕ(aw) = ϕ(a) + ϕ(w) forany a ∈ A , w ∈ A ∗.Language: L = {w ∈ A ∗ | ϕ(w) = 0 mod 5}

Genus and size

Another example.

Genus and size

Another example.

Genus and size

Another example.

Genus and size

Topological size

Definition

The topological size of a language L is

|L|top = min{|A| | L(A) = L, g(A) = g(L)}.

By definition: |L|top ≥ |L|.

The topological size |L|top is regarded as “the cost” you are willingto pay for the simplest topological embedding of the representingautomaton of L.

Genus and size

Topological size

Definition

Genus and size

Topological size

Definition

Genus and size

Question

Is there a universal bound |L|top ≤ f (|L|) for some explicit functionf ?

If such a function exists, it has to be at least exponential.

Theorem (2015)

There is a family of planar regular languages (Ln)n≥1 such that forsome K > 2, |Ln|top = O(K |Ln|).

Genus and size

Question

Is there a universal bound |L|top ≤ f (|L|) for some explicit functionf ?

If such a function exists, it has to be at least exponential.

Theorem (2015)

There is a family of planar regular languages (Ln)n≥1 such that forsome K > 2, |Ln|top = O(K |Ln|).

Genus and size

Book and Chandra (1978) raised the question of whether theplanarity of a language is decidable.

One may generalize the question and ask whether the following istrue.

Conjecture

The genus of a regular language is computable.

Partial positive answer:

Theorem (2012, 2015)

If the language has “no short cycles”, the conjecture is true.

Genus and size

Book and Chandra (1978) raised the question of whether theplanarity of a language is decidable.One may generalize the question and ask whether the following istrue.

Conjecture

Theorem (2012, 2015)

Genus and size

Conjecture

Theorem (2012, 2015)

Genus and size

Conjecture

Theorem (2012, 2015)

Genus and size

“No short cycles”.

Definition

A language has no cycles of length less than k if the underlyinggraph of its minimal automaton has no cycles of length less than k .

Cycle = simple cycle = closed path without repeated edge (nomatter its orientation), regardless of the orientation of the originalgraph.

Theorem (2012)

Let L be a language on m letters. Assume that m ≥ 4 and that Lhas no cycles of length ≤ 2. Then

1 +m − 3

6|L| ≤ g(L) ≤ 1 +

m − 1

Genus and size

Definition

Theorem (2012)

1 +m − 3

6|L| ≤ g(L) ≤ 1 +

m − 1

Genus and size

Definition

Theorem (2012)

1 +m − 3

6|L| ≤ g(L) ≤ 1 +

m − 1

Genus and size

Hierarchies of languages

Remark Every language on one letter is planar (exercise).

Book and Chandra (1978) construct an example of a language ontwo letters which is nonplanar from a minimal deterministicautomaton with 35 states.

Theorem (2012, 2015)

Let A be an alphabet of at most 2 letters. There exists a family oflanguages (Ln)n∈N on alphabet A such that g(Ln) = n.

Remark. The minimal example of nonplanar language on twoletters has 30 states. Conjecture. 30 is optimal.

Genus and size

Theorem (2012, 2015)

Genus and size

Theorem (2012, 2015)

Genus and size

Theorem (2012, 2015)

Remark. The minimal example of nonplanar language on twoletters has 30 states.

Conjecture. 30 is optimal.

Genus and size

Theorem (2012, 2015)

Genus and size

The following definition is the “directed version” of Fellows’ graphemulator (in connection with the planar finite cover conjecture inthe 1980s).

Definition

Let G = (E ,V ) be a directed graph. A directed emulator of G is agraph G = (E , V ) such that there is a surjective simplicial mapϕ : G → G sending surjectively outgoing edges of each vertexv ∈ V onto outgoing edges of the image vertex ϕ(v).

Genus and size

The following definition is the “directed version” of Fellows’ graphemulator (in connection with the planar finite cover conjecture inthe 1980s).

Definition

Let G = (E ,V ) be a directed graph. A directed emulator of G is agraph G = (E , V ) such that there is a surjective simplicial mapϕ : G → G sending surjectively outgoing edges of each vertexv ∈ V onto outgoing edges of the image vertex ϕ(v).

Genus and size

Idea: the directed emulator map mimicks the canonical projectionmap between an automaton and its minimal automaton.

Theorem

A language L has genus ≤ g iff (the underlying directed graph of)its minimal automaton Amin has a directed emulator of genus ≤ g.

This leads to a ”directed minor” approach to the computation ofthe genus of a language as an analogy to the Robertson-Seymourtheorem for graphs.

More on this: come to Denis Kuperberg’s talk.

Genus and size

Theorem

Genus and size

Theorem

This leads to a ”directed minor” approach to the computation ofthe genus of a language

as an analogy to the Robertson-Seymourtheorem for graphs.

Genus and size

Theorem

Genus and size

Theorem

Genus and size

The genus of regular languages is only the tip of the iceberg. Manyother invariants inspired from low-dimensional topology and graphtheory admit nontrivial reincarnations in the study of languages.

Chromaticity : noStar height: has a topological refinement. Existence of a hierarchy.Computability: unknown.Graph width, cohomology theories based on graphs...

Genus and size

The genus of regular languages is only the tip of the iceberg. Manyother invariants inspired from low-dimensional topology and graphtheory admit nontrivial reincarnations in the study of languages.Chromaticity : no

Star height: has a topological refinement. Existence of a hierarchy.Computability: unknown.Graph width, cohomology theories based on graphs...

Genus and size

The genus of regular languages is only the tip of the iceberg. Manyother invariants inspired from low-dimensional topology and graphtheory admit nontrivial reincarnations in the study of languages.Chromaticity : noStar height: has a topological refinement. Existence of a hierarchy.Computability: unknown.

Graph width, cohomology theories based on graphs...

Genus and size

The genus of regular languages is only the tip of the iceberg. Manyother invariants inspired from low-dimensional topology and graphtheory admit nontrivial reincarnations in the study of languages.Chromaticity : noStar height: has a topological refinement. Existence of a hierarchy.Computability: unknown.Graph width, cohomology theories based on graphs...

the genus of regular languages and other ideas from low ...deloup/topcomp2016/slides/genus.pdf ·...

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